Curves and Surfaces (I) 姜明 北京大学数学科学学院 Based on [EA], Chapter 11. 更新时间2019年4月29日星期一10时22分10秒

Introduction In computer graphics, flat objects are popular in this virtual world Graphic systems can render them at high rates. We need methods to model curved objects.

Outline Representation of Curves and Surfaces Design Criteria Parametric Cubic Polynomial Curves Cubic Interpolating Polynomial

Representation of Curves and Surfaces Three major ways of object representation Explicit Representation Implicit Representations Parametric Form Polynomial Representations Parametric Polynomial Curves Parametric Polynomial Surfaces

Explicit Representation No guarantee that this representation exists for a given curve/surface z=f(x,y) cannot represent a (full) sphere. Some curves and surfaces may not have an explicit representation. Coordinate-system-dependent effect. Easy to obtain points on them.

Implicit Representations f(x,y,z) = 0 in 3D or f(x,y) = 0 in 2D. Less coordinate-system-dependent: it does represent all lines/circles. Allow to determine whether points lie on the curve/surface. Difficult to find points on the curve/surface. Curves in 3D are not easily represented in implicit form because it takes two equations to represent a curve in 3D f(x,y,z) = 0 and g (x,y,z) = 0 “In general, most of the curves and surfaces that arise in real applications have implicit representations. Their use is limited by the difficult in obtaining points on them.” [EA, p. 600] Algebraic surfaces: f(x,y,z) is a polynomial. Of particular importance are the quadric surfaces.

Parametric Form Same form in 2D and 3D. Most flexible and robust for computer graphics than the others. Still coordinate-system-dependent. Coordinate-system-independent representations are possible Using Frenet frame for curves. Difficult to determine if a point is on the curve/surface.

Parametric Polynomial Curves/Surfaces Parametric representations are not unique. Parametric polynomial forms are of most use in computer graphics. Curve Segment Surface Patch

Design Criteria Local control of shape: A single global description is generally out of the question and too complex. We would like/have to work interactively with the shape, carefully molding it meet specifications Smoothness and continuity at joint points A curve with discontinuity is of little interest to us. Not only will each segment have to be smooth, but also we want a degree of smoothness where the segments meet at joint point. Smoothness is measured with derivatives along the curves/surfaces. Ability to evaluate derivatives is needed to evaluate smoothness and normals. Stability It is necessary to bend curves/surfaces to the desired shape through local control points. When we make a change, this change will only affect the shape in only the area where we are working. Small changes in the values of input parameters should cause only small changes in output variables We are usually satisfied if the curve/surface passes close to the control points. Ease of Rendering Good math representations may be of limited value if they cannot be efficiently rendered.

Splines Splines are types of curves, originally developed for ship-building in the days before computer modeling. Naval architects needed a way to draw a smooth curve through a set of points. The solution was to place metal weights (called knots) at the control points, and bend a thin metal or wooden beam (called a spline) through the weights. The physics of the bending spline meant that the influence of each weight was greatest at the point of contact, and decreased smoothly further along the spline. To get more control over a certain region of the spline, the draftsman simply added more weights. This scheme had obvious problems with data exchange! People needed a mathematical way to describe the shape of the curve. Cubic Polynomials Splines are the mathematical equivalent of the draftsman's wooden beam. Polynomials were extended to B-splines (for Basis splines), which are sums of lower-level polynomial splines. Then B-splines were extended to create a mathematical representation called NURBS. Read History of Splines Edited from http://www.alias.com/eng/support/studiotools/documentation/Using/About4.html

Parametric Cubic Polynomial Curves We must choose the degree of polynomials. If we choose higher degree We will have more parameters that we can set to form the desired shape. The evaluation of points will be costly. There is more danger that the curve will become rougher at some points. If we pick too low a degree, we may not have enough parameters to work and to make them to join smoothly. If we design each curve segment over a short interval, we can achieve many of our purposes with low-degree curves. Cubic polynomials are chosen, at least initially. c is to be determined from the control-point data. Types of cubic curves differ in how they use the control-point data. The data may be Interpolating: the polynomial must pass some points. Higher order interpolating at certain parameter values. Smoothness conditions at join points. Approximative: the curve pass close to some data points.

Cubic Interpolating Polynomial Suppose that we have four control points in 3D. We seek the coefficients c such that the polynomial p(u) = uTc passes through the four points. We assume the interval is [0, 1], the curve passes four points at u = 0, 1/3, 2/3, and 1.

Joining of interpolating segments. Because of the lack of derivative continuity at joint points, the interpolating polynomial is of limited use in computer graphics. Joining of interpolating segments.

Blending Functions

Cubic Interpolating Patch A bicubic surface patch can be written in the form Suppose that we have 16 3D control points pij. We can use those points to define an interpolating bicubic surface patch at interpolating points 0, 1/3, 2/3, 1. It is easy to find the interpolating polynomial by tensor product.